Archie Equation

Gustavus E. Archie (1942)

The Cornerstone of Quantitative Log Interpretation.

Water Saturation (Sw) Relate to:

Formation Resistivity (Rt)
Formation Porosity
Formation Water Resistivity (Rw)
Parameters a-m-n
Archie's Equation

Electrical Conduction in Clean, Water-bearing Rock

The most influential paper in petrophysics

" The Electrical Log as an aid in Determining
Some Reservoir Characteristics" (1942)

Before 1942,
(1) Determing HC Reserves was difficult
and expensive, because the only reliable
method was to core the formation using oil-base mud and measure
Sw in Lab.

(2) Electrical Logs were used for identify HC-bearing zone, not used for
quantitative evaluation.
Sw Definition

Water Saturation (Sw) is

the Fraction (Percentage) of Pore Volume
of the Reservoir Rock that is
Filled with Water

This Water Includes Free & Bound Water

Pore space may contain Oil, Gas and Water
cfc10-3301.42 200x
Saturation

Formation saturation is defined as the fraction of its pore volume (porosity) occupied by a
given fluid.

Saturation =Volume of a specific fluid

pore volume

Definitions
Sw = water saturation.
So = oil saturation.
Sg = gas saturation.
Sh = hydrocarbon saturation = So + Sg

Saturations are expressed as percentages or

fractions, e.g.
Water saturation of 75% in a reservoir with
porosity of 20% contains water equivalent to
15% of its volume.
Sw: Water Saturation

Valid or Invalid depend on:

(1) Rt measurement/correction
(2) Porosity measurement/derivation (Company policy)

(3) V-shale determination (Company policy)

(4) A-M-N derivation (Company policy)

(5) Saturation Model (Company policy)

(6) Rw derivation (Company policy)

Water Saturation

Using well log data are based on equations that make use
Sw Calculation
"RATIO OF RESISTIVITY"

1. Electrical current flow is essentially

through the water in the rock 's pore

2. Oil, gas and solid rock do not conduct electrical

Assumptions current (Insulators)

3. If H.C. replace some pore water, the rock's

resistance will increase
(If Rt > Ro imply the presence of HC.)

1. Low Rt Pay Sands

Exceptions 2. Any deceptive Rt measurements
 Rw
Formation Water Resistivity

Resistivity of Formation water

Determine in the clean water bearing zone

Problems:
(1) Representative or not
(2) Variations

Very Critical in Sw calculation

Ro

(1) Resistivity of 100% Water Bearing zone

Rock + 100% water
(2) Ro > Rw

(3) In clean water bearing zone

Rt = Ro
Rw----Ro

Rw Current
path

Unit volume filled with only water

Current path

Ro

Unit volume with water and matrix

Rt Rt---> RLLd, RILd
Measured by the tool

Typical Formation Rock+Water+HC

Oil
Water
Sand grain

Grain surface water

Current path
Rt-Normal Case

- Electrical Conduction resulted from the transport of ions in the pore-filling

brine predominantly NaCl

- How easily these ions traverse a rock's pore system determing the rock's
resistivity

- High porosity with open, well-connected pores have low reistivity

- Low porosity with sinuous, constricted pores system, have high reistivity

- Hydrocarbon further blocks the ions' paths and increases a rock's

reistivity

Archie's equation
n quantify the relationship of Sw with
porosity and resistivity
Rt-Special Cases

(1) Isolated (Vuggy) porosity, more resistive

(2) Fractures, tend to lower resistivity

Because they provided straight paths for ions

(3) Shaly Rocks: shale lower reistivity by

cation conduction

(4) Conductive minerals- pyrite

Electrical conduction through the rock grains
dramatically lower the resistivity
 Rock containing pores saturated
with water and hydrocarbons

Non-shaly rock, 100% saturated with water

having resistivity, Rw

Rt
Cube of water having
φ= 20%
Sw = 20%
resistivity, Rw
SHC =80%
Ro
φ= 20%
Sw = 100%
R es
istiv Rw
ity
φ= 100%
Sw = 100%
(1) Rock
Conductivity
Increasing
Resistivity
Increasing

(2) Gas
(3) Oil
a
(4) Fresh Water
F = Ro =
(5) Salt Water Rw φm
 Resistivity---General Introduction (3)
Most rock materials are essentially insulators

The enclosed fluids are conductors

Hydrocarbons are exception to fluisd conductivity
Hydrocarbons are infinitely resistive
Formation=Matrix+Fluid
The primitive assumption for HC exploration:
When a formation is porous and contains salty water---Resistivity is low
When the same formation contains hydrocarbons, Resistivity is high
High resistivity values may indicate a porous, hydrocarbon
bearing formation

GR Rt

Rt

OWC
Ro
Archie Equation (1st relationship)

Formation Factor:

Ro Rock  Water (100% )

F 
Rw Formation Water
Non-shaly rock, 100% saturated
with water having resistivity, Rw

Cube of water
having resistivity,
Rw
Ro
= 20%
Sw = 100%

Rw
= 100%
Sw = 100%

Formation Factor
Equation
Formation Factor
 -R..

Bri ＂﹒

．．φ－m
一， A., • FR,.
﹒口國 R,.

-R， ＝ 間。s.."

Ol 砂Brlne Satu同’，ed Raeκ

three formations which have the same
porosity but different values of formation
resistivity factor, F.

The role of the matrix is evident: less at low

values of F﹛top), greater at high values of F
(bottom).
Formation Factor

• The formation factor (F) depends on

– Porosity of the formation,
– Pore geometry - tortuosity
– Lithology of the formation
– Degree of cementation, and
– Type and amount of clay in the rock
Formation Factor

Rock grain structure

Pore geometry(tortuosity)

Porosity
 Formation Factor - Example Core Data

a  1.0
Archie 1/5/96
 Formation Factor

Rock grain structure The relationship generate a m

Pore geometry(tortuosity) factor

Cementation exponent

Porosity
Formation Factor Equation
• Archie’s equation for formation factor relate to Ro, Rw, and Porosity

F=R0/Rw=a·-m 1000
Rock type 1

100
F

10 Rock type 2

1
Note: Sw=1 .01 .1 1.0
 From NExT, 1999
Formation Factor vs. Porosity---->m

• F=a·-m
a = constant  1.0 for most formations
m = cementation factor  2 for most formations

• Other commonly used values

– Sandstones:
• F = 0.8/2 (Tixier)
• 0.62/2.15 (Humble)

– Carbonates
• F = 0.8/2
 Formation Factor

Ro=F x Rw

Describes the effect of the presence of the rock matrix.

F = 1.00 for a rock with 100% porosity, i.e., no matrix, just 100% fluid.
Formation factor has no units because it is the ratio of two resistivities.
In real rocks F takes values usually between 20 and 500.
Formation factor is related to the porosity of the rock and the connectivity of the pore
spaces (effect of the tortuous pathways).

The cementation index is the factor that describes the increase in resistivity that results
from the insulating mineral grains forcing the current to take tortuous pathways through
the conducting fluid.
Both the formation factor and the cementation exponent can be measured on core plugs
in the laboratory.
 Cementation Exponent

In real rocks the cementation index usually varies between 1.0 and 3.0

Values between 1.0 and 1.4 are associated with igneous and metamorphic rocks that contain
fractures. Fractures are a form of porosity that is localized and well connected, and hence
approximates to the situation where we had uniform tubes of porosity going through the sample.

Values between 1.4 and 2.0 are found in sandstones, with the higher values found in
more consolidated sandstones, where the current flow paths are more tortuous.

Values between 2.0 and 2.6 are typical for carbonates, and represent a greater degree of
tortuosity in the current flow that is found in carbonates because much of the porosity in
carbonates is unconnected (e.g., vugs).

In general, the value of the cementation exponent increases as the degree of connectedness of
the pore network diminishes, which rather supports it being called the cementation exponent.
Archie Equation----- 1st relationship

Formation Factor and Porosity

1 a
F m  m
 
Archie equation 2nd relationship

Relationship of

Ro Rt
 Resistivity---General Introduction (3)
Most rock materials are essentially insulators

The enclosed fluids are conductors

Hydrocarbons are exception to fluisd conductivity
Hydrocarbons are infinitely resistive
Formation=Matrix+Fluid
The primitive assumption for HC exploration:
When a formation is porous and contains salty water---Resistivity is low
When the same formation contains hydrocarbons, Resistivity is high
High resistivity values may indicate a porous, hydrocarbon
bearing formation

GR Rt

Rt

OWC
Ro
Rock containing pores saturated with
water and hydrocarbons

What is the relationship of Rt, Ro with

fully and partially saturated HC ? w
i
Rt t
φ= 20%
Sw = 20% h
SHC =80%

Ro
R es φ= 20%
istiv
ity Sw = 100%

Clean rock 100% saturated with

(1) Rock water having resistivity, Rw
Conductivity
Increasing
Resistivity
Increasing

(2) Gas
(3) Oil
a
(4) Fresh Water F = Ro =
(5) Salt Water Rw φm
Archie equation 2nd relationship

Define the relationship of Rt and Ro as

Resistivity Index

Rt
Resistivity Index=
Ro
 Resistivity---General Introduction (3)
Most rock materials are essentially insulators

The enclosed fluids are conductors

Hydrocarbons are exception to fluisd conductivity
Hydrocarbons are infinitely resistive
Formation=Matrix+Fluid
The primitive assumption for HC exploration:
When a formation is porous and contains salty water---Resistivity is low
When the same formation contains hydrocarbons, Resistivity is high
High resistivity values may indicate a porous, hydrocarbon
bearing formation

GR Rt

Sw= ?
Rt
Sw= ?

OWC
Sw=100% Ro

There are must have relation between RI and

water saturation
Archie 1/5/96
Resistivity Index vs. Sw
Saturation exponent

It is common to hear that, in the absence of laboratory measurements, the

saturation exponent has been taken to be equal to 2.

Its exact value may vary depending on the wettability of the rock.
Oil-wet system usually have higher valuue ( >2, may up to 4~5)
Some water-wet system show n may less than 2.

To get n by measurement and Archie equation.

 Resistivity Index
Partial Water Saturation

Rt=I*Ro

resistivity index and describes the effect of partial desaturation of the rock.
If the rock is fully saturated, I=1.00.
If the rock is full of dry air (i.e., not saturated with a conductive fluid).
The resistivity index therefore varies between unity and infinity depending upon
the degree of saturation of the rock.
The saturation exponent normally has a range of values from 1.8 to 2.0, however much
lower and much higher values have been found.
The value of the saturation exponent can be obtained from laboratory experiments on
core samples. The procedure is as follows for a single core sample.
Archie Equation derivation

Rt
Resistivity Index=
Ro
Water Saturation and Resistivity:

Rt 1 n
 n  Sw Sw=1 all water in the
Ro Sw Rt=Ro
Sw=0
pore
all HC in the pore
Rt=infinite

n Ro Boundary
condition satisfy
Regardless of n

Sw 
Rt
Archie Equation

Formation water
Formation water
+Rock
Ro  F  Rw
Ro 1 a
F  m  m
Rw Ro Rw  
a
 Ro  m  Rw


Rt  I  Ro
1 Rt
I 
Sw n Ro
Formation water
+Rock+H.C. Rt n a  Rw
 Sw 
 m  Rt
Archie Equation derivation

Ro a a
F  m Ro  Rw 
Rw  m

Rt 1 Ro
IR   n Sw n

Ro Sw Rt

Ro a 1 a  Rw
Sw n
  Rw  m   m
Rt  Rt   Rt
Sw n a
 m  Rw
  Rt
 Generalized Archie Equation

n a  Rw
Sw 
m  Rt

Practical Equation

a  Rw n=2
Sw  m=2

2
  Rt a=0.81 for consolidated sand
a=0.62 for unconsolidated sand
a=1 for carbonate
 a  Rw
Archie Equation Sw n 
 m  Rt

a m n
Tortuosity factor Cementation factor Saturation exponent
m higher--> Sw higher
m lower --> Sw lower
a=0.62 unconsolidated ss
more compacted/cemented n higher--> Sw higher
a=0.81 consolidated ss
---> higher m n lower --> Sw lower
a=1.0 carbonate
unconsolidated ss
---> lower m

Rw porosity Rt
Rw higher--> Sw higher Por. higher--> Sw lower
Rt higher -->Sw lower
Rw lower --> Sw lower Por. lower --> Sw higher
Rt lower --> Sw higher
Rw from catalogs, SP, or Por. from Den., Neu, BHC
Rt from ILd, LLd after corrected
samplings or cross plots
Summary of G.E. Archie (1942) Clean Sand Model

(1) For clean sands, Archie suggested:

The resistivity of brine-saturated rock, Ro, was related brine resistivity Rw
Ro=F*Rw
F called formation factor, depend only on porosity
F=1/Porm
Archie determined m=2
(2) Archie believed the electrical conduction was entirely
due to ionic transport in the brine
(3) For H.C. bearing sands, Archie proposed Rt is equal Ro multiplying
a factor, the resistivity index I
Rt=Ro*I ( I=Rt/Ro)
(4) Previous researchers' data led him to propose:
n
SW =1/I
n=saturation exponent , was about 2
Parameter Source

Rt Deep investigation resistivity tool

From SP log
Rw Calculated from water zone
Measured on RFT sample
Sonic tool
Por. Formation density tool
Neutron tool"

m Measured in laboratory
Guessed"

Measured in laboratory
n Guessed
 Concerns on Archie Equation

(1) Most Pratical Log Interpretation today are Based on.

(2) Archie Equation derived for clean , water-wet sand
(3) More than 50 different shaly sand equations, frequently
are variations of Archie equation.

(4) No equation is definitive or universally accepted.

(5) Archie equation may not hold for some cases:
(a) Non-unimodal pore-size distribution
(b) Conductive mineral content
(c) Vuggy or moldic porosity
(d) Fracture and microporosity

Archie-2 1/5/96
 Archie Equation
Application

1. F-Overlay

2. Hingle Plot

3. Pickett Plot

4. Rwa
Petrophysics MSc Course Notes Resistivity Theory

17.8 The Hingle Plot

Theory. This plot is based on Eq. (17.18). The objective is to obtain a linear cross-plot of the Rt data
measured by the resistivity tool and the φ data measured by one of the porosity tools. To do this all of
the exponents of Eq. (17.18) are multiplied by –1/m to give

Rt1 / m  Rw1 / m φ S w n / m
(17.20)

Because in any given reservoir we can take Rw, m and n as constant, and because we will apply the
equation for selected values of Sw, Eq. (17.20) becomes

Rt1 / m  Bφ
(17.21)

where B is a constant. The Hingle graph paper is designed such that the y-axis represents Rt-1/m so that
Rt can be entered directly in the plot. This implies that a different form of graph paper is needed for
each value of m. The x-axis on the Hingle grid is porosity on a linear scale.

Figure 17.10 shows a Hingle plot.

Application. The use of the Hingle plot is as follows. For any given reservoir zone carry out the
following steps:

 Construct the 100% water saturation (Sw=1) line. The first point on this line is automatically
available, as Rt is infinite when φ=0, and this point plots in the bottom left hand corner of the
Hingle grid. The second point is calculated with knowledge of Rw for the reservoir. Equation
(17.11) is used to calculate Ro knowing Rw for the reservoir, for the value of m relevant to the
Hingle grid, and at any value of φ (the higher the better for accuracy). For example, in Fig. 17.10,
the m value is 2, and if Rw=0.4 ohm.m, we can say that at the arbitrary porosity of φ =0.2, the value
of Ro=10 ohm.m. The Ro, φ point can be plotted on the grid and joined with the first point by a
straight line. This is the water line, and represents how Ro varies with porosity when the rock is
fully saturated with water.

 Other lines for partial water saturations can now be constructed. Their first point is always in the
bottom left hand corner of the Hingle grid because Rt is always infinite when φ=0 no matter what
the water saturation. The second point is calculated from Eq. (17.17) at a given arbitrary porosity
assuming or knowing the value of n and calculating Rt from the relevant Ro, which is available
from the water line. For a particular partial saturation line (Sw=0.5, say) the Rt, φ point can be
plotted on the grid and joined with the first point by a straight line. This is the Sw=0.5 line, and
represents how Rt varies with porosity when the rock is 50% saturated with water.

 A fan of partial saturation lines can be constructed in this way, say for every 10% increment in
water saturation. A large number of porosity and Rt pairs are now extracted from the logs and
plotted on the graph. It is immediately obvious how much water saturation is present on average,
and the water saturation for particular points (relating to a particular depth) can be estimated from
the graph by interpolation between the iso-saturation lines.

Dr. Paul Glover Page 213

Petrophysics MSc Course Notes Resistivity Theory

17.8 The Hingle Plot

Theory. This plot is based on Eq. (17.18). The objective is to obtain a linear cross-plot of the Rt data
measured by the resistivity tool and the φ data measured by one of the porosity tools. To do this all of
the exponents of Eq. (17.18) are multiplied by –1/m to give

Rt1 / m  Rw1 / m φ S w n / m
(17.20)

Because in any given reservoir we can take Rw, m and n as constant, and because we will apply the
equation for selected values of Sw, Eq. (17.20) becomes

Rt1 / m  Bφ
(17.21)

where B is a constant. The Hingle graph paper is designed such that the y-axis represents Rt-1/m so that
Rt can be entered directly in the plot. This implies that a different form of graph paper is needed for
each value of m. The x-axis on the Hingle grid is porosity on a linear scale.

Figure 17.10 shows a Hingle plot.

Application. The use of the Hingle plot is as follows. For any given reservoir zone carry out the
following steps:

 Construct the 100% water saturation (Sw=1) line. The first point on this line is automatically
available, as Rt is infinite when φ=0, and this point plots in the bottom left hand corner of the
Hingle grid. The second point is calculated with knowledge of Rw for the reservoir. Equation
(17.11) is used to calculate Ro knowing Rw for the reservoir, for the value of m relevant to the
Hingle grid, and at any value of φ (the higher the better for accuracy). For example, in Fig. 17.10,
the m value is 2, and if Rw=0.4 ohm.m, we can say that at the arbitrary porosity of φ =0.2, the value
of Ro=10 ohm.m. The Ro, φ point can be plotted on the grid and joined with the first point by a
straight line. This is the water line, and represents how Ro varies with porosity when the rock is
fully saturated with water.

 Other lines for partial water saturations can now be constructed. Their first point is always in the
bottom left hand corner of the Hingle grid because Rt is always infinite when φ=0 no matter what
the water saturation. The second point is calculated from Eq. (17.17) at a given arbitrary porosity
assuming or knowing the value of n and calculating Rt from the relevant Ro, which is available
from the water line. For a particular partial saturation line (Sw=0.5, say) the Rt, φ point can be
plotted on the grid and joined with the first point by a straight line. This is the Sw=0.5 line, and
represents how Rt varies with porosity when the rock is 50% saturated with water.

 A fan of partial saturation lines can be constructed in this way, say for every 10% increment in
water saturation. A large number of porosity and Rt pairs are now extracted from the logs and
plotted on the graph. It is immediately obvious how much water saturation is present on average,
and the water saturation for particular points (relating to a particular depth) can be estimated from
the graph by interpolation between the iso-saturation lines.

Dr. Paul Glover Page 213

Petrophysics MSc Course Notes Resistivity Theory

Figure 17.10 The Hingle plot.

Dr. Paul Glover Page 214

Petrophysics MSc Course Notes Resistivity Theory

17.9 The Pickett Plot

Theory. The Pickett Plot is also based on Eq. (17.18).

In a water-bearing formation we can write from Eq. (17.11)

Ro  Rw φ m (17.22)

which, when rearranged becomes

log Ro  log Rw  m log φ (17.23)

So a plot of log Ro against log  gives a straight line. The value on the y-axis is equal to log Ro when
φ=1, and the slope of the line is –m.

In a hydrocarbon-bearing formation we can write from Eqs. (17.11) and (17.15)

Rt  I Ro  I Rw φ m (17.24)

which, when rearranged, becomes

log Rt  log I  log Rw  m log φ (17.25)

which is the same straight line as described by Eq. (17.23), with the same gradient, but with a parallel
shift equal to log I.

Application. The Pickett Plot plots the formation resistivity Rt against the porosity on a log-log scale.
The data form straight lines with a gradient equal to –m. Hence, the cementation exponent can be
calculated. If one has data in the water zone of the reservoir, Eqs. (17.22) and (17.23) hold true, and
the value on the y-axis when the line intersects φ=1, gives log Rw from which Rw can be calculated.
The line is called the water line.

If one has data in the oil-bearing zone, and the value of Rw is known, the value on the y-axis when the
line intersects φ=1, gives log I + log Rw from which I can be calculated if Rw is known. If the saturation
exponent is then known, we can use the I value to calculate the water saturation.

Alternatively, we can establish the water line and construct iso-saturation lines with the same gradient
that are offset from the water line by values of log I that represent increments in water saturation.
Plotting the formation resistivity and porosity values from logs on this plot then allows the mean
saturation in the reservoir to be judged, and particular values of water saturation at a given depth can
be calculated can be approximated by interpolation between the iso-saturation lines.

Figure 17.11 shows a Pickett plot example.

Dr. Paul Glover Page 215

Petrophysics MSc Course Notes Resistivity Theory

17.9 The Pickett Plot

Theory. The Pickett Plot is also based on Eq. (17.18).

In a water-bearing formation we can write from Eq. (17.11)

Ro  Rw φ m (17.22)

which, when rearranged becomes

log Ro  log Rw  m log φ (17.23)

So a plot of log Ro against log  gives a straight line. The value on the y-axis is equal to log Ro when
φ=1, and the slope of the line is –m.

In a hydrocarbon-bearing formation we can write from Eqs. (17.11) and (17.15)

Rt  I Ro  I Rw φ m (17.24)

which, when rearranged, becomes

log Rt  log I  log Rw  m log φ (17.25)

which is the same straight line as described by Eq. (17.23), with the same gradient, but with a parallel
shift equal to log I.

Application. The Pickett Plot plots the formation resistivity Rt against the porosity on a log-log scale.
The data form straight lines with a gradient equal to –m. Hence, the cementation exponent can be
calculated. If one has data in the water zone of the reservoir, Eqs. (17.22) and (17.23) hold true, and
the value on the y-axis when the line intersects φ=1, gives log Rw from which Rw can be calculated.
The line is called the water line.

If one has data in the oil-bearing zone, and the value of Rw is known, the value on the y-axis when the
line intersects φ=1, gives log I + log Rw from which I can be calculated if Rw is known. If the saturation
exponent is then known, we can use the I value to calculate the water saturation.

Alternatively, we can establish the water line and construct iso-saturation lines with the same gradient
that are offset from the water line by values of log I that represent increments in water saturation.
Plotting the formation resistivity and porosity values from logs on this plot then allows the mean
saturation in the reservoir to be judged, and particular values of water saturation at a given depth can
be calculated can be approximated by interpolation between the iso-saturation lines.

Figure 17.11 shows a Pickett plot example.

Dr. Paul Glover Page 215

Petrophysics MSc Course Notes Resistivity Theory

Figure 17.11 The Pickett Plot.

Dr. Paul Glover Page 216

Appendix:
The Electrical Resistivity Log as an Aid in Determining Some
Reservoir Characteristics
BY G. E. ARCHIE(1942)
The Electrical Resistivity Log as an Aid in Determining Some
Reservoir Characteristics
BY G. E. ARCHIE*
﹛Dallas Meeting, Oct。，ber I94I)
THE usefulness of the electrical resistivity log the resistivity of the mud in the borehole,
in determining reservoir characteristics is the effect of invasion of the mud filtrate
governed largely by: (1) the accuracy with into the formation, the relation of the
which the true resistivity of the formation can
recorded thickness of beds to electrode
be determined ﹔（ 2) the scope of detailed data
spacing, the heterogeneity of geologic
concerning the relation of resistivity measure­
ments to formation characteristics﹔（3) the formations, the salinity or conductivity
available information concerning the conduc­ of connate water, and, perhaps of greatest
tivity of connate or formation waters ﹔（4) the importance, the lack of data indicating the
extent of geologic knowledge regarding proba­ relationship of the resistivity of a formation
ble changes in facies within given horizons, both in situ to its character and fluid content.
vertically and laterally, particularly in relation On the Gulf Coast it is found that the
to the resultant effect on the electrical proper­ e宜ects of the size of the borehole and the
ties of the reservoir. Simple examples are given mud resistivity are generally of little
in the following pages to illustrate the use of
importance, except when dealing with
resistivity logs in the solution of some problems
dealing with oil and gas reservoirs. From the high formational resistivities or extremely
available information, it is apparent that much low mud resistivities. Fortunately, little
care must be exercised in applying to more practical significance need be attached to
complicated cases the methods suggested. It the exact values of the higher resistivities
should be remembered that the equations given recorded. Low mud resistivities are not
are not precise and represent only approximate common, but when this condition is
relationships. It is believed, however, that encountered it may be corrected by
under favorable conditions their application replacing the mud column. With• the
falls within useful limits of accuracy.
present advanced knowledge of mud
INTRODUCTION control, invasion of mud filtrate into
The electrical log has been used exten­ sands can be minimized, thereby increasing
sively in a qualitative way to correlate the dependability of the electrical log.
formations penetrated by the drill in the The effect of electrode spacing on the
exploitation of oil and gas reservoirs and recorded thickness of a bed is often subject
to provide some indication of reservoir to compensation or can be su血ciently
content. However, its use in a quantitative accounted for to provide an acceptable
way has been limited because of various approximation of the true resistivity of
factors that tend to obscure the signi益cance the formation. As development of a且eld
of the electrical readings obtained. Some or area progressively enhances the knowl­
of these factors are the borehole size, edge of the lithologic section, the resistivity
values of the electrical log take on greater
Manuscrip_t received at the 。但ce of the Institu社 e
Sept. 27﹔revised Dec. 8, I94I Issued as T.P. 1422 in significance, ultimately affording accept­
。.， 。
PETROLEUM TECHNOLOGY, January z942.
* Shell Oil C H uston; Texas. able interpretations. The salinity, and

54
 G. K. ARCHIE 55
therefore the conductivity, of the connate per liter. The following simple relation
water associated with the various produc- was found to exist for that range of
ing horizons may be determined with porosities and salinities:
sufficient accuracy by the usual sampling
procedure. R. = FR", [I]
Determination of the significance of where R. = resistivity of the sand when
the resistivity of a producing formation all the pores were filled with brine, R", =
as recorded by the electrical log appears, resistivity of the brine, and F = a "for-
for the present at least, to rest largely mation resistivity factor."
with the application of empirical relation- In Figs. I and 2, F is plotted against
ships established in the laboratory between the permeabilities and porosities, respec-
certain of the physical properties of a tively, of the samples investigated. The
reservoir rock and what may be termed data presented in Fig. I were obtained
a formation factor. It should be stressed from consolidated sandstone cores in
at this point that numerous detailed which the cementing medium consisted
laboratory studies of the physical proper- of various amounts of calcareous as well
ties of the formations in relation to the as siliceous materials. The cores had
electrical measurements in question are essentially the same permeability, parallel
essential to a reliable solution of the to and perpendicular to the bedding of
problems dealing with reservoir content. the layers. All of the cores were from
The purpose of this paper is to present producing zones in the Gulf Coast region.
some of these laboratory data and to Cores from the following fields were used:
suggest their application to quantitative Southeast Premont, Tom Graham, Big
studies of the electrical log. It is not in- Dome-Hardin, Magnet-Withers, and Sheri-
[ended to attempt to discuss individual dan, Texas; also La Pice, and Happy town,
resistivity curves and their application. La. Fig. 2 presents similar data obtained
The disturbing factors (borehole, bed from cores of a widely different sandstone;
thickness, and invasion) are discussed that is, one that had extremely low per-
briefly only to indicate instances when meability values compared with those
they are not likely to affect the usefulness shown in Fig. I for corresponding porosities.
of the observed resistivity. These cores were from the Nacatoch
sand in the Bellevue area, Louisiana.
RESISTIVITY OF SANDS WHEN PORES ARE
From Figs. I and 2 it appears that the
ENTIRELY FILLED WITH BRINE
formation resistivity factor F is a function
A study of the resistivity of formations of the type and character of the formation,
when all the pores are filled with water and varies, among other properties, with
is of basic importance in the detection of the porosity and permeability of the reser-
oil or gas by the use of an electrical log. voir rock; many points depart from the
Unless this value is known, the added average line shown, which represents a
resistivity due to oil or gas in a formation reasonable relationship. Therefore, indi-
cannot be determined. vidual determinations from any particular
The resistivities of a large number of core sample may deviate considerably
brine-saturated cores from various sand from the average. This is particularly
formations were determined in the labora- true for the indicated relationship to
tory; the porosity of the samples ranged permeability. Further, although the varia-
from 10 to 40 per cent. The salinity of the tion of F with porosity for the two groups
electrolyte filling the pores ranged from of data taken from sands of widely different
20,000 to 100,000 milligrams of NaCI character is quite consistent, the effect
56 ELECTRICAL RESISTIVITY LOG AND RESERVOIR CHARACTERISTICS

of variations in permeability on this ity. Thus, knowing the porosity of the

factor is not so evident. Naturally the sand in question, a fair estimate may be
two relationships could not be held to made of the proper value to be assigned
apply with equal rigor because of the well to F, based upon the indicated empirical
100

I.. 50 ,
.g
u
....
IS

'"
-+-
.:;;
t; 10
.;;;
-
"
~
x

x
x ~
x'

~
x x~
\ 'K
~
5 5
~
E
L-
o
LL.
I
I 5 10 50 100 500 1000 5000 0.10 0.30 1.00
PermeOibility, milliolarcys Porosity
FIG. I.-RELATION OF POROSITY AND PERMEABILITY TO FORMATION RESISTIVITY FACTOR FOR CON-
SOLIDATED SANDSTONE CORES OF THE GULF COAST.

500

'\0

.
o •
0
0
0
0

0
0

. ..
.
0

0
0
-
oJ' ... 0
0
~
\ I

f\,

I
0.1 0.5 1.0 5 10 50 100 0.10 0.30 1.00
Permeability. millidClrcys Porosity
FIG. 2.-RELATION OF POROSlTY AND PERMEABILITY TO FORMATION RESISTIVITY FACTOR, NACATOCH
SAND, BELLEVUE, LA.
Permeabilities below o. I millidarcynot recorded.

established fact that permeability does not relationship

bear the same relation to porosity in all F = 8- m
sands. From close inspection of these data, or from Eq. I,
R. = R,.fJ-m
and at the present stage of the investiga-
tion, it would appear reasonably accurate to where 8 is the porosity fraction of the
accept the indicated relationship between sand and m is the slope of the line represent-
t he formation resistivity factor and poros- ing the relationship under discussion.
 G. E. ARCHIE 57
From a study of many groups of data, m which the oil or gas is distributed in the
has been found to range between 1.8 pores may be so different that these
and 2.0 for consolidated sandstones. For relations derived in the laboratory might
clean unconsolidated sands packed in not apply underground.
the laboratory, the value of m appears
to be about 1.3. It may be expected, <:
o
:;=
then, that the loosely or partly consolidated E
sands of the Gulf Coast might have a ::J

~ O.301--------"oo.;~~-----_l
value of m anywhere between 1.3 and 2. L

RESISTIVITY OF FORMATIONS WHEN PORES ~I,

ARE PARTLY FILLED WITH BRINE, THE CI.) O.IO'-------~------""'~
I 10
REMAINING VOIDS BEING FILLED WITH R Resistivity of oil Or g~s sond
Ro Resistivity of some sond 100 per cent w~ter·boorinq
OIL OR GAS
R
Various investigators-Martin, 1 Jako- FIG. S·-RELATION OF S TO R.
sky,2 Wyckoff,3 and Leverett 4-have stud- Legend and Data
ied the variation in the resistivity of sands
Salinity
due to the percentage of water contained Type of Water. Oil Porosity
Curve Investi- Grams or Frac-
in the pores. This was done by displacing gator Sand NaCI per Gas tion
varying amounts of conducting water Liter

from the water-saturated sand with non- -- --- --- ---

- - Wyckoff Various CO. Various
conducting fluid. Fig. 3 shows the relation - - Martin
Leverett Uncons. 8 approx. Oil 0·40
which the various investigators found to --- Cores 130 Oil 0.20 and
0.45(?)
exist between S (fraction of the voids --I Jakosky Friable 29 approx. Oil 0.23

filled with water) and R (the resulting

resistivity of the sand) plotted on loga- Considerable encouragement on this
rithmic coordinates. For water saturations point is established, however. For example,
down to about 0.15 or 0.20, the following Eq. 4 appears to hold even though gas or
approximate equation applies: oil is the nonconducting phase. Each
1 probably assumes a different distribution
S= (~y or R = R.s-" [4] in the pores, yet the resulting resistivity
is not appreciably changed. Also, no great
For clean unconsolidated sand and for change is found in the average relation
consolidated sands, the value of n appears between the formation resistivity factor
to be close to 2, so an approximate relation and porosity for changes in types of con-
can be written: solidated sandstones. This indicates that
even though the oil or gas underground
S=~ [51 may fill the pore space in a different
or from Eq. I,
manner from that in the short-time

S= ~F;w [61
laboratory experiments, the relationship
expressed by Eq. 4 should apply equally
well underground.
Since in the laboratory extremely short
BASIC RESISTIVITY VALUES TO BE OBTAINEE
intervals of time were allowed for the
IN ESTIMATING FLUID CONTENT OF A SANE
establishment of the equilibrium conditions
compared with underground reservoirs, The foregoing discussion indicates that
there is a possibility that the manner in the basic values to be obtained are: (I) tht
I References are at the end of the paper. resistivity of the sand in question under·
58 ELECTRICAL RESISTIVITY LOG AND RESERVOIR CHARACTERISTICS

ground (R), and (2) the resistivity of the Consider a borehole penetrating a
same sand when its pores are entirely large homogeneous layer, in which case
filled with connate water (R.). the electrode spacing is small in comparison
The first value can be obtained from the with the thickness of the layer. If the
electrical log when all factors can be resistivity of the mud in the hole is the
properly weighed. The latter may also be same as the resistivity of the layer, there
obtained from the log when a log is avail- will be, of course, no correction for the
able on the same horizon where it is entirely effect of the borehole. If the resistivity
water-bearing. Of course, this is true only of the mud differs from the resistivity of
when the sand conditions, particularly por- the layer, there will be a correction.
osity, are the same as at the point in ques- Table 1 shows approximately how the
tion and when the salinity of the connate presence of the borehole changes the
or formation water throughout the horizon observed resistivity for various conditions.
is the same. The third curve, or long normal, of the
In a water-drive reservoir, or any Gulf Coast is considered because this
reservoir where the connate water is in arrangement of electrodes gives very
direct contact with the bottom or edge nearly a symmetrical picture on passing
water, there should be no appreciable a resistive layer and has sufficient pene-
difference in the salinities through the tration in most instances to be little
horizon, at least within the limits set forth affected by invasion when the filtrate
for the operation of Eqs. 1 and 4; that is, properties of the mud are suitable.
when the salinity of the connate water
is over 20,000 mg. NaCl per liter and the TABLE I.-E.ffect of Borehole on Infinitely
connate water is over 0.15. In depletion- Large Homogeneous Formation
type reservoirs, or when connate water Observed Resisttvity on ffiectric Log
In an 8-in. In a Is-in.
is not in direct contact with bottom or Borehole Borehole
True
edge water, special means may have to Resistivity
Resisti vity of
Mud in Hole ~~~tf:iilol;
be devised to ascertain the salinity of the of Formation, (at Bottom-hole (at Bottom-hole
Meter-ohms Temperature) of Temperature) of
connate water. 0·5 1.5 0·5 1.5
Meter- Meter- Meter- Meter-
When it is not possible to obtain R. ohms ohms ohms ohms
in the manner described above, the value --- --- ---
0.5 0·5 0·5 0·5 0·5
can be approximated from Eq. 3, () and m I I I I I
5 6 5 5 5
having been determined by core analyses 10 I. II II II

and R .. by regular analyses. 50 65 65

I 50 55

CALCULATION OF CONNATE WATER, POROS- The values in Table I have been cal-
lTY AND SALINITY OF FORMATION WATER culated assuming a point potential "pick-
FROM THE ELECTRICAL LOG up" electrode 3 ft. away from a point
The resistivity scale used by the electrical source of current, other electrodes assumed
logging companies is calculated assuming to be at infinity, and it has been found
the electrodes to be points in a homo- that the table checks reasonably well
geneous bed. 5 Therefore, the values re- with field observations. Checks were
corded must be corrected for the presence made by: (I) measuring the resistivity of
of the borehole, thickness of the layers shale and other cores whose fluid content
in relation to the electrode spacing, and does not change during the coring operation
any other condition different from the and extraction from the well; (2) measuring
ideal assumptions used in calculating the the resistivity of porous cores from water-
scale. bearing formations after these cores were
 G. E. ARCHIE 59
resaturated with the original formation tivities. I(is assumed that large shale bodies
water. Adjustment due to temperature are present above and below the beds, at
difference, of course, is necessary before the same time neglecting the presence of
the laboratory measurement is compared the borehole and again assuming point
with the field measurement. electrodes.

Resistivity •meter· ohms

TABLE 2.-EiJect of Formation Thickness,
No Borehole Present O~~~IO~;;:;;;;~200
-~~~----;-~--,--~3480
True Resistivity Observed Resistivity
Layer between Large Thickness of Layer --~~-----+-Q-_+--__I3500
Shale Bodies Having
Resistivity of 1.0
Meter-ohms 24 Ft. 16 Ft. 8 Ft.
--- --- --- ---r=-------I----.O~"i='......,......d. 3520
NormOiI curve,
1 1 I 1
5 5 5 3 -+-------+-~_++._-__I3540
10 10 9 6 ~'i.
20 20 19 II ~
-~-~----~_+-~~m~~'--~3560
Long normal ". 't'~)
curve --~~~::~~~, $';
The correction at the higher resistivities -----(;=---------+--"~~=---I3580

appears to be appreciable. However,

in the Gulf Coast when the value of R. =-----+-;;;~==l3600
is low the correction is not so important.
- ; - - - - - - - + 7..'"-r"'-=--t--------l36Z0
For example, assume a friable oil sand
whose true resistivity is 50 meter-ohms --------'-----'-------'3640
and whose resistivity when entirely water- FIG. 4.-ELECTRICAL LOG OF AN EAST TEXAS
bearing is 0.50 meter-ohms; the connate WELL.
Diameter of hole, i% in.; mud resistivity,
water would occupy about 0.10 of the 3.4 at 85°F.; bottom-hole temperature, approxi-
pore volume CEq. 5). However, if the mately 13SoF.
observed value on the log, 65 meter-
ohms, were used without correcting for Tables I and 2 assume ideal conditions,
the borehole, the connate water would be so if the sand is not uniform, or if invasion
calculated to occupy 0.09 of the pore affects the third curve, the observed re-
volume. Therefore, although the effect sistivity values may deviate farther from
of the borehole size and mud resistivity the true value. The magnitude of the
on the observed resistivity readings may influencing factors, of course, willlim,it the
be appreciable, the resultant effect on usefulness of the observed resistivity value
the calculated connate-water content of recorded on the log. Invasion of the mud
the sand is not important. filtrate is probably the most serious factor;
When the thickness of the formation however, as previously mentioned, it can
is very large in comparison with the often be controlled by conditioning the
electrode spacing, there will, of course, be mud flush for low filtrate loss.
no correction to make for the thickness Fig. 4 shows a log of an East Texas well.
of the layer. However, when the thickness The observed resistivity on the long normal
of the formation approaches the electrode curve for the interval 3530 to 3560 ft. is
spacing, the observed resistivity may be 62 meter-ohms, or, from Table I, approxi-
very different from the true value. Table 2 mately 50 meter· ohms after correcting for
shows approximately what the third curve the borehole. In this instance the mud
(long normal) of the Gulf Coast would resistivity at the bottom-hole temperature
read for certain bed thicknesses and resis- of 135°F. is approximately 2.2 meter-ohms.
60 ELECTRICAL RESISTIVITY LOG AND RESERVOIR CHARACTERISTICS

The interval is thick enough so that there volume. The accepted value assigned for
should be no appreciable effect due to the connate-water content of the East
electrode spacing. The formation is more or Texas reservoir is 17 per cent.
less a clean friable sandstone, so Eq. 5 can An electrical log of a sand in the East
White Point field, Texas, is shown in Fig. 5.
Resistivity. The observed resistivity at 4075 ft. is
meter-ohms
Self-pofenf/al 0 5 10 approximately 5 meter-ohms. The value of
--':""--:;}---.+--r---,4040 F for this sand by laboratory determination
---I 25 J+-t
mv. is 6. The sand is loosely consolidated, hav-
ing 32 per cent porosity average. The
- - - j - - - - j r - - - 4 l i : - t " - - 4060 resistivity of the formation water by direct
measurement is 0.063 meter-ohms at the
bottom-hole temperature of 138°F. There-
fore, R. = 6 X 0.063 or 0.38 meter-ohms.
-*=------+-"Zlrt---14080 This checks well with the value obtained
Normal curve- --> I
by the electrical log between the depths of
4100 and 4120 ft., which is 0.40 (see
amplified third curve). Therefore, invasion
probably is not seriously affecting the
third curve. From Tables I and 2 it appears
that the borehole and electrode spacing do
not seriously aff~ct the observed resistivity
FIG. 5.-ELECTRICAL LOG OF A SAND IN EAST at 4075 ft. The connate water is approxi-
WHITE. POINT FIELD, TEXAS.
0'38 or 0.27.
Diameter of hole, 7% in.; mud resistivity,
1.7 at 80°F.; bottom-hole temperature, 138°F. mately ~--,
5.0

Other uses of the empirical relations may

be used to approximate the connate-water have occurred to the reader. One would be
content. The formation resistivity factor
the possibility of approximating the maxi-
for this sand is approximately IS, using
mum resistivity that the invaded zone
Eq. 2 where 8 = 0.25 and m = 1.8. The
could reach (wh!!n formation water has a
resistivity of the formation water by actual greater salinity than borehole mud) by
measurement is 0.075 meter-ohms at a Eq. I, where R", would now be the resistiv-
bottom-hole temperature of 135°F. There- ity of the mud filtrate at the temperature of
fore, from Eq. I, R. for this sand is the formation and F the resistivity factor
IS X 0.075 = 1.1 meter-ohms. This value of the formation near the borehole. By
checks reasonably well with the value knowing the maximum value of resistivity
recorded at 3623 to 3638 ft. on this log as that the invaded zone could reach, the
well as on the many logs from this pool limits of usefulness of the log could be
where the Woodbine sand is water-bearing; better judged. For example, assume that a
i.e., 0.9 to 1.5 meter-ohms. The close check porous sand having an F factor of less than
obtained between the calculated and re- IS was under consideration. If the mud
corded resistivity of the water sand indi- filtrate resistivity were 0.5 meter-ohms, the
cates that invasion is not seriously affecting resistivity of the invaded zone, if com-
the third curve. Solving Eq. 5, the connate pletely flushed, would be IS X 0·5 = 7.5.
water of the zone 3530 to 3560 ft. occupies Thus the observed resistivity values of this
a.pproximately § = 0.15 of the pore sand up to approximately 7.5 meter-ohms
'\j50 could be due to invasion.
 DISCUSSION

ing tl,le electrodes the apparent resistance

ACKNOWLEDGMENT measurement is uniquely determined from the
Cooperation of the Shell Oil Co., Inc., specific resistance and position of each and all
and permission to publish this paper are of the particles making up the sphere. Any
gratefully acknowledged. The resistivity rational interpretation of these apparent resist-
ance measurements is possible only for the
measurements on the numerous cores were
simplest combinations of particles and their
performed under the supervision of S. H. specific resistances. Fortunately, soils, sub-
Rockwood and J. H. McQuown, of the soils and subsurface rocks, with their embodied
Shell Production Laboratories. fluids and gases, vary greatly in this property
among themselves. For example, clay appears
REFERENCES to have an average specific resistance of
I. Martin, Murray and Gillingham: G.ophysics (July approximately 50 to. 150 foot-ohms, whereas
1938).
2. Jakosky and Hopper: Geophysics {Jan. 1937). for sand and gravel the specific resistance is
3. Wyckoff and Botset: Physics (Sept. 1936) 325. roughly from 2000 to 5000 foot-ohms. The
4. Leverett: Trans. A.I.M.E. (1938) J32, 149.
S. C. and M. Schlumberger and Leonardon: Trans. important feature is the great absolute differ-
A.I.M.E. (1934) 110, 237.
ences in resistance, consequently a resistance
profile across a buried lens of sand or gravel sur-
DISCUSSION rounded by clay produces a striking response.
(H. F. Beardmore presiding) In spite of the amount of control available
and the freedom for selecting various electrode
S. W. WILCOX,* Tulsa, Okla.-This paper intervals, no reliable quantitative predictions
recalls some of my own observations on the could be made that were not related to bound-
correlation of the electrical resistance of earth ary surfaces. The probable depth to the first
materials with their other physical properties. discontinuity-namely, the clay-sand contact
While Geophysical Engineer for the Depart- -could be determined fairly accurately if the
ment of Highways, of the State of Minnesota, thickness of the sand body was considerable.
from 1933 until 1936, I was primarily engaged When the depth to the sand was known from
in conducting earth-resistivity surveys pros- independent data, or could be assumed to be
pecting for and exploring sand and gravel constant, it was possible to predict its thick-
deposits. This work was done by two field ness. If both were known, a good guess might
parties using equipment of the Gish-Rooney be made regarding the depth to the water·
type, and was carried out in every part of the table; and, in addition, if all these were known,
state, both winter and summer. a surmise could be made about the quality of
In brief, when a sand or gravel prospect was the sand; i.e., whether it contained organic
discovered, in any way, it was detailed by the material or was weathered. Perhaps if the
·resistivity survey to outline its extent and to degrees of control were sufficient the porosity of
locate test holes for field and laboratory sample the sand, its grain size, or even its temperature
analysis. This survey consisted of a grid of might be predicted.
"steptraverses" of one or more electrode I observed that few of these variables, even
separations, and for each an "iso-ohm," or the ones that generally contribute to the bulk
equal resistance contour plan map, was drawn. of the readings, could be quantitatively
Several thousand earth-resistivity readings separated without additional independent data;
were taken over more than one hundred therefore my interpretation was necessarily
prospects. In some instances the test pitting empirical and based on experience. Fortun-
was started before the completion of electrical ately, in sand and gravel prospecting the
survey and their findings were soon available economically most important factors contribute
for checking any suspected correlation theory their effects in the same direction. A high
and confirming what subsurface factors were apparent resistance indicates either a thin body
being measured and how effectively. of highly resistant gravel near the surface, or a
From accepted earth-resistivity theory, it thicker one overlain with more clay stripping.
follows that within a definite sphere surround- Clean gravel is more resistant than weathered,
* Seismograph Service Corporation. and hard gravel more so than soft.
62 ELECTRICAL RESISTIVITY LOG AND RESERVOIR CHARACTERISTICS

In practical terms, I found that an apparent My interpretation problems appeared to be

resistance reading of 500 foot-ohms for a essentially similar to those of electrical well
20-ft. electrode separation recorded over logging where the operator, after observing
ground or glacial moraines of southern Minne- the character of the resistance and the self-
sota reliably suggested a deposit of sand or potential curves, tells his client whether pipe
gravel worth further investigation. As a matter should be set. The accuracy of his prediction is
of record, prospecting in the part of the state based largely on experience and not on slide-
where these materials are very scarce, less than rule calculations.
3 per cent of the test holes located on the Mr. Archie's paper suggests an experimental
geophysical information failed to yield granular attack for expanding and improving the
materials of commercial quality and quantity interpretation technique of electrical well
for at least highway subgrade treatment. Jogging. Any contribution of this nature that
Varying the electrode interval gave additional increases its effectiveness is of great value to
confirmation as to the thickness of the deposit the petroleum industry. I offer my own experi-
and very little else. ences and observations to emphasize that he
In connection with our field work, we made has tackled a difficult research problem and
extensive laboratory studies, attempting to wish him luck.
work out the relation between the moisture
content of sand and gravel and its specific
resistance. These apparently simple eXlleri- Dr. A. G. LOOMIS, * Emeryville, Calif.··-In
ments were not of much help in clearing up my the laboratory, we take into account the varia-
field interpretations. Several variables were tions in measured resistivities of sands and tap
very hard to control in the laboratory. water by finding out the cause of the variations
The analogy between this type of earth- in resistivity. That is, if the tap water itself
resistivity mapping and electrologging is close. varied from day to day, its electrolyte content
The first measures electrical impedance along a must vary from day to day and chemical
surface generally parallel to the bedding planes; analysis would indicate the change. If sands
the latter, up a borehole more or less perpen- did not give consistent resistivity readings, the
dicular to them. The same general limitations character of the sands (in other words, the
and possibilities appear to be common to both formation resistivity factor) probably changed
methods. Obviously, controls for checking are or the kind and amount of water contained in
easier to obtain for plan mapping than for the sand must have varied.
well logging within the depth of effective
penetration. * Shell Development Co.